Figure 1: Two-state trajectories

A two-state trajectory (also termed two-state time trajectory or a trajectory with two states) is a dynamical signal that fluctuates between two distinct values: ON and OFF, open and closed, ${\displaystyle +/-}$, etc. Mathematically, the signal ${\displaystyle X(t)}$ has, for every ${\displaystyle t,}$ either the value ${\displaystyle X(t)=c_{\mathrm {off} }}$ or ${\displaystyle X(t)=c_{\mathrm {on} }}$.

In most applications, the signal is stochastic; nevertheless, it can have deterministic ON-OFF components. A completely deterministic two-state trajectory is a square wave. There are many ways one can create a two-state signal, e.g. flipping a coin repeatedly.

A stochastic two-state trajectory is among the simplest stochastic processes. Extensions include: three-state trajectories, higher discrete state trajectories, and continuous trajectories in any dimension.^[1]

Two state trajectories in biophysics, and related fields

Two state trajectories are very common. Here, we focus on relevant trajectories in scientific experiments: these are seen in measurements in chemistry, physics, and the biophysics of individual molecules^[2][3] (e.g. measurements of protein dynamics and DNA and RNA dynamics,^{[4][5][6][7][8]} activity of ion channels,^[9][10] enzyme activity,^{[11][12][13][14][15]} quantum dots^{[16][17][18][19][20][21]}). From these experiments, one aims at finding the correct model explaining the measured process.^{[22][23][24][25][26][27][28][29][30][31][32]} We explain about various relevant systems in what follows.

See also

• Single-molecule experiment
• Reduced dimensions form
• Kinetic scheme
• Master equation
• Wave

References